Coloring the Dth Power of The Cartesian Product of Two Cycles and Two Paths

Abstract

The d th power graph G d is defined on the vertex set of a graph G in such a way that distinct vertices with distance at most d in G are joined by an edge. In this paper the chromatic number of the d th power of the Cartesian product C m C n of two cycles is studied and some of the exact value of χ((C m C n) d) with conditions are determined. Also the chromatic number of the d th power of grid P m P n with some conditions are determined and the exact value of χ((P m P n) d) for n = 2, 3 is obtained.